# Gaussian Distribution Formula

Gaussian distribution is a very common in a continuous probability distribution. The Gaussian distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables.

The probability density function formula for Gaussian distribution is given by,

$\large f(x,\mu , \sigma )=\frac{1}{\sigma \sqrt{2\pi}}\; e^{\frac{-(x- mu)^{2}}{2\sigma ^{2}}}$

Where,
$x$ is the variable
$\mu$ is the mean
$\sigma$ is the standard deviation

### Solved Examples

Question 1: Calculate the probability density function of Gaussian distribution using the following data. x = 2, $\mu$ = 5 and $\sigma$ = 3

Solution:

From the question it is given that,
x = 2, $\mu$ = 5 and $\sigma$ = 3

Probability density function formula of Gaussian distribution is,

f(x, $\mu$, $\sigma$ ) = $\frac{1}{\sigma \sqrt{2\pi }}$$\;$$e^{\frac{-(x-\mu )^{2}}{2\sigma ^{2}}}$

f(2, 5, 3 ) = $\frac{1}{\sigma \sqrt{2\pi }}$$e^{\frac{-(2-5 )^{2}}{18}}$ = 0.1330$\times$2.2182 = 0.2950

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